How do I rescale an object to be a specified cm in length and have the rest of the object scale to accommodate?
I don't think there's a way of doing it directly.

I do it by:


CtrlA > Applying the scale of the object
Adjusting the known dimension in the 'Dimensions' panel
Copy and pasting the scale from the known dimension to the others in the 'Scale' panel
(CtrlA > Applying the scale of the object, again)


EDIT: 2.8 makes one way a bit easier...


Select the known dimension in the 'Scale' panel
Right-click and select 'Copy as Driver'
Right-click in the other scale fields and 'Paste Driver'


Now the known dimension will drive the other 2, when you adjust the known Dimension.